Cubic B‐Spline Collocation Method for One‐Dimensional Heat and Advection‐Diffusion Equations

Abstract

Numerical solutions of one-dimensional heat and advection-diffusion equations are obtained by collocation method based on cubicB-spline. Usual finite difference scheme is used for time and space integrations. CubicB-spline is applied as interpolation function. The stability analysis of the scheme is examined by the Von Neumann approach. The efficiency of the method is illustrated by some test problems. The numerical results are found to be in good agreement with the exact solution.

Numerical Method Using Cubic Trigonometric B-Spline Technique for Nonclassical Diffusion Problems

“…The study of B-spline functions is a key element in computer-aided geometric design [32][33][34][35]. It has also attracted attention in the literature [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51] to the numerical solution of various differential equations [38][39][40]. This is because they have important geometric properties and features that make them amenable to more detailed analysis.…”

“…Numerical methods based on B-spline functions of various degrees have been utilized for solving initial and boundary value problems. As examples, a cubic B-spline collocation method was used to solve a nonlinear diffusion equation subject to certain initial and Dirichlet boundary constraints [41], a finite element method based on bivariate splines has been used for solving parabolic partial differential equation [42], and the combination of finite difference approach and cubic B-spline method was applied for the solution of a one-dimensional heat equation subject to local boundary constraints [43,44]. Goh et al [45] presented a comparison of cubic B-spline and extended cubic uniform B-spline based collocation methods for solving a one-dimensional heat equation with a nonlocal initial constraint and concluded that extended cubic uniform B-spline with an appropriate value of parameters gives better results than the cubic B-spline.…”

“…Goh et al [45] presented a comparison of cubic B-spline and extended cubic uniform B-spline based collocation methods for solving a one-dimensional heat equation with a nonlocal initial constraint and concluded that extended cubic uniform B-spline with an appropriate value of parameters gives better results than the cubic B-spline. A finite difference scheme based on cubic B-spline was also used for solving the one-dimensional wave equation [43], advection-diffusion equation [44], one-dimensional coupled viscous Burgers' equation [47], system of strongly coupled reaction-diffusion equations [48], and one-dimensional hyperbolic problems [49].…”

A Semi-Lagrangian Scheme for Advection-Diffusion Equation

“…Cubic spline polynomials can be written in the form ( ) = N + N ( − N ) + N ( − N ) . + N ( − N ) \ , N ≤ ≤ NGA (10) Where N is the concentration value of at node and N , N , N are polynomial coefficients determined from values at nodes. The detailed description about construction of the cubic spline polynomials and calculation of coefficients in the polynomials are given, for example, at [20].…”

“…However, since this equation contains two different physical processes such as advection and diffusion, the precise numerical solution is quite difficult. To overcome this difficulty such as classical finite difference method [4], high-order finite element method [5], high-order finite difference methods [6,7], green element method [8], cubic and extended Bspline collocation methods [9][10][11], cubic, quartic and quintic B-spline differential quadrature methods [12,13], method of characteristics unified with splines [14][15][16], cubic trigonometric B-spline approach [17] Taylor collocation and Taylor-Galerkin methods [18] , Lattice Boltzmann method [19] have been developed. In addition, with the help of operator splitting methods, the appropriate methods for the physical processes of the problem can be combined.…”